Given that most furnace transformers are 40 VA and 24 V, 40/24 Amps is about 2 A. So, 3A is not a bad rating to use at all for a contact. The contact could basically switch a shorted relay winding and have the fuse blow with no damage to the relays.
40 VA was enough, and power to spare to activate a fan, heat relay and gas valve or a fan and compressor relay.
The OP is building a thermostat or should we say multiple thermostats in a zoned system and wants something small and cheap.
So, let's revisit the original problem:
A small and cheap replacement for a 24 VAC relay for use in a typical heating/cooling thermostat where the intention is building a thermostat from scratch.
I'd look at the OPTOMOS series of relays:
https://www.digikey.com/product-search/en/relays/solid-state-relays/1048664?k=optomos
Now, they could be a little power hungry, but some require currents on the order of a mA to switch, so one might not be able to pass the power budget.
Call 4 batteries, $5.00 and the device to add a wire $25.00. Payback is 5 years. Without batteries (OK, maybe a lithium or supercap), the unit is smaller, but you won't have heat if you don't have an extra set of batteries. More savings could result if you could run the extra wire in some places.